MATLAB does not have derivatives built in so let\'s build our own numerical firs

Question

MATLAB does not have derivatives built in so let's build our own numerical first and second lerivative functions. Write a MATLAB function called LastNamemyderivs that takes as input parameters (/, n,xo,c), where n = 1 or n = 2, and returns the 0(h'x) approximation for with (accuracy. Similar to problems 5 & 6 on Homework #4, your program should start with h = 0.5 in the 0(h^4) finite difference formulas, and successively halve h until two successive iterates are within (of one another; it would then return the later iterate as the approximation to message if n is neither 1 nor 2. Write a MATLAB function, railed LastNamenewtonf. that implements Newton's method. Your should take as parameters (f, f) where:rn is the initial eu ess. Your Newton iteration

Explanation / Answer

ANSWER b

syms x y; % Make x,y symbolic
f1 = x^2 + y^3 - 1; % Make your two equations (from your example)
f2 = x^4 - y^4 + x*y;
f = [f1; f2]; % f(x) vector

% Jacobian matrix
J = [diff(f1, 'x') diff(f1, 'y'); diff(f2, 'x') diff(f2, 'y')];

% Initial vector
x0 = [1; 1];

% Tolerance:
tol = 1e-10;

Now, make your script into a function:

% To run in MATLAB, do:
% [n, xout, tol] = LastNamenewtonf(f, J, x0, tol);
% disp('n = '); disp(n); disp('x = '); disp(xout); disp('tol = '); disp(tol);

function [n, xout, tol] = LastNamenewtonf(f, J, x0, tol)

% Just to be sure...
syms x, y;

% Initialize error
ep = 1; % Note: eps is a reserved keyword in MATLAB

% Initialize counter
n = 0;

% For the beginning of the loop
% Must transpose into a row vector as this is required by subs
xout = x0';

% Computation loop   
while ep > tol && n < 100
g = subs(f, {x,y}, xout); %g(x)
ep = abs(g(1)) + abs(g(2)); %error   
Jg = subs(J, {x,y}, xout); %LastNamenewtonf
yout = xout - Jgg; %iterate   
xout = yout; %update x   
n = n + 1; %counter+1   
end

% Transpose and convert back to number representation
xout = double(xout');

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.